Simulation 1

Data structure: \(O = (W, A, Y)\)

  • U - exogenous variables
  • W - baseline covariate that is a measure of body condition
  • A - treatment level based on W, continuous between 0 and 5
  • Y - outcome, indicator of an event

Underlying data generating process, \(P_{U,X}\)

  • Exogenous variables:
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
    • \(U_Y \sim Uniform(min = 0, max = 1)\)
  • Structural equations F and endogenous variables:
    • \(W = U_W\)
    • \(A = bound(2 - 0.5W + U_A, min=0, max=5)\)
    • \(Y = \mathbf{I}[U_Y < expit(-5 + W + 2.25A -0.5WA)]\)

Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve

{r child = '2_1_Simu_results_template.Rmd'} #

Simulation 2

Data structure: \(O = (W, A, Y)\)

  • U - exogenous variables
  • W - baseline covariate that is a measure of body condition
  • A - treatment level based on W, continuous between 0 and 5
  • Y - outcome, indicator of an event

Underlying data generating process, \(P_{U,X}\)

  • Exogenous variables:
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 1.3^2)\)
    • \(U_Y \sim Uniform(min = 0, max = 1)\)
  • Structural equations F and endogenous variables:
    • \(W = U_W\)
    • \(A = bound(2.5 - 0.5W + U_A, min=0, max=5)\)
    • \(Y = \mathbf{I}[U_Y < expit(-8 + 2W + 3sin(1.5A^{1.5}) + 2WA)]\)

Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve

{r child = '2_1_Simu_results_template.Rmd'} #

Simulation 3

Data structure: \(O = (W, A, Y)\)

  • U - exogenous variables
  • W - baseline covariate that is a measure of body condition
  • A - treatment level based on W, continuous between 0 and 5
  • Y - outcome, indicator of an event

Underlying data generating process, \(P_{U,X}\)

  • Exogenous variables:
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
    • \(U_Y \sim Uniform(min = 0, max = 1)\)
  • Structural equations F and endogenous variables:
    • \(W = U_W\)
    • \(A = bound(2 - 0.5W + U_A, min=0, max=5)\)
    • \(Y = \mathbf{I}[U_Y < expit(-10 - 3W + 4A + \mathbf{I}(A>2) * 5sin((0.8A)^2 - 2.6) )]\)

Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve

{r child = '2_1_Simu_results_template.Rmd'} #

Simulation 4

Data structure: \(O = (W_1, W_2, W_3, W_4, W_5, A, Y)\)

  • W - baseline covariates
  • A - treatment level based on W, continuous between 0 and 5
  • Y - outcome, indicator of an event

Underlying data generating process, \(P_{U,X}\)

  • Exogenous variables:
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
  • Structural equations F and endogenous variables:
    • \(W \sim N(\mu_W, \Sigma_W)\)
    • \(A = bound(0.1W_1 + 0.2W_2 + 0.5W_3 + 0.15W_4 - 0.05W_5 - 0.01W_3W_5 + U_A, min=0, max=5)\)
    • \(Y = \mathbf{I}[U_Y < expit(-7 -W_1 + 2W_2 - 0.5W_4 - W_1W_3 + 4A + 0.5AW_2)]\)
      • where \(\mu_W = \begin{bmatrix}0 \\0 \\5 \\1 \\1 \\\end{bmatrix}\), and \(\Sigma_W = \begin{bmatrix}1&0.8&1.35&0.1&0.03 \\0.8&1&1.2&0.15&0.03 \\1.35&1.2&2.25&0.075&0.225 \\0.1&0.15&0.075&0.25&0.1125 \\0.03&0.03&0.225&0.1125&0.09 \\\end{bmatrix}\)

Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve

{r child = '2_1_Simu_results_template_0.Rmd'} #

Simulation 5

Data structure: \(O = (W_1, W_2, W_3, W_4, W_5, A, Y)\)

  • W - baseline covariates
  • A - treatment level based on W, continuous between 0 and 5
  • Y - outcome, indicator of an event

Underlying data generating process, \(P_{U,X}\)

  • Exogenous variables:
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
  • Structural equations F and endogenous variables:
    • \(W \sim N(\mu_W, \Sigma_W)\)
    • \(A = bound(0.1W_1 + 0.2W_2 + 0.5W_3 + 0.15W_4 - 0.05W_5 - 0.01W_3W_5 + U_A, min=0, max=5)\)
    • \(Y = \mathbf{I}[U_Y < expit(-10 -W_1 + 2W_2 - 0.5W_4 - 0.5W_1W_3 + 4A + \mathbf{I}(A>2) *5sin((0.8A)^2 - 2.6))]\)
      • where \(\mu_W = \begin{bmatrix}0 \\0 \\5 \\1 \\1 \\\end{bmatrix}\), and \(\Sigma_W = \begin{bmatrix}1&0.8&1.35&0.1&0.03 \\0.8&1&1.2&0.15&0.03 \\1.35&1.2&2.25&0.075&0.225 \\0.1&0.15&0.075&0.25&0.1125 \\0.03&0.03&0.225&0.1125&0.09 \\\end{bmatrix}\)

Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve

{r child = '2_1_Simu_results_template_0.Rmd'} #

Simulation 6

Data structure: \(O = (W_1, W_2, W_3, A, Y)\)

  • W - baseline covariates
  • A - treatment level based on W, continuous between 0 and 1
  • Y - outcome, indicator of an event

Underlying data generating process, \(P_{U,X}\)

  • Structural equations F and endogenous variables:
    • \(W_1 \sim Uniform(o,1)\)
    • \(W_2 \sim Bernoulli(\mu=0, \sigma^2 = 2^2)\)
    • \(W_3 \sim N(W_1, 0.25*exp(2W_1))\)
    • \(A \sim Beta(v(W)\mu(W), v(W)[1-\mu(W)])\)
    • \(Y \sim Bernoulli(Q_0(A,W))\)
      • where:
      • \(v(W) = exp(1 + 2W_1expit(W3))\)
      • \(\mu(W) = expit(0.03 - 0.8log(1+W_2) + 0.9exp(W_1)W_2 - 0.4arctan(W_3+2)W_2W_1)\)
      • \(\bar{Q}_0(A,W) = expit(-2 + 1.5A + 5A^3 - 2.5W_1 + 0.5AW_2 - log(A)W_1W_2 + 0.5A^{3/4}W_1W_3)\)

Outcome of interest: \(E_0[Y|a,W]\), \(a \in (0,1])\), the causal dose-response curve

{r child = '2_1_Simu_results_template_s6.Rmd'} #

Simulation 7

Data structure: \(O = (W, A, Y)\)

  • W - baseline covariates
  • A - treatment level based on W, continuous between 0 and 1
  • Y - outcome, indicator of an event

Underlying data generating process, \(P_{U,X}\)

  • Exogenous variables:
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
    • \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
    • \(U_Y \sim Uniform(min = 0, max = 1)\)
  • Structural equations F and endogenous variables:
    • \(W = U_W\)
    • \(A = bound(2.5 - 0.5W + U_A, min=0, max=5)\)
    • \(Y = \mathbf{I}[U_Y < expit(-6 + W + 3.5A\mathbf{I}(A\geq2) - 4A\mathbf{I}(A\geq4) - 0.5WA)]\)

Outcome of interest: \(E_0[Y|a,W]\), \(a \in [0,5]\), the causal dose-response curve

##        W                   A               Y         
##  Min.   :-3.781093   Min.   :0.000   Min.   :0.0000  
##  1st Qu.:-0.687019   1st Qu.:1.750   1st Qu.:0.0000  
##  Median :-0.006363   Median :2.479   Median :1.0000  
##  Mean   :-0.011178   Mean   :2.492   Mean   :0.5489  
##  3rd Qu.: 0.664196   3rd Qu.:3.243   3rd Qu.:1.0000  
##  Max.   : 3.730121   Max.   :5.000   Max.   :1.0000
## Summary of A given W < -1:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   2.493   3.278   3.220   3.945   5.000
## Summary of A given -1 < W <= 0:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   1.993   2.691   2.677   3.342   5.000
## Summary of A given 0 < W <= 1:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   1.610   2.284   2.292   2.946   5.000
## Summary of A given 1 < W:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   1.024   1.766   1.760   2.447   5.000

n = 200

CV vs Undersmoothing (Zero smoothness order with default number of knots)

##  The average lambda of CV-HAL: 0.0052 (= 1 * lambda_CV )
##  The average lambda of globally undersmoothed HAL: 0.0004 (= 0.0681 * lambda_CV )  The average lambda of globally undersmoothed HAL: 0.0004 (= 0.0681 * lambda_CV )  The average lambda of globally undersmoothed HAL: 0.0004 (= 0.0681 * lambda_CV )
##  The average number of bases of CV-HAL: 10.3024   The average number of bases of CV-HAL: 10.3024
##  The average number of bases of globally undersmoothed HAL: 36.8775
##  The average fitting time for CV-HAL: 0.9538 seconds
##  The average fitting time for globally undersmoothed HAL: 1.4834 seconds

Undersmoothed Smoothness-adaptive HAL

##    smooth_order if_n_knots_default sl_pick lambda lambda_scaler n_basis cv_risk
## 1         0.000                  1  0.0000 0.0036        1.0000   6.908  0.3904
## 12        0.000                  0  0.0780 0.0047        1.0000  11.148  0.2258
## 14        0.000                  0  0.0780 0.0047        1.0000  11.148  0.2258
## 15        0.000                  0  0.0780 0.0047        1.0000  11.148  0.2258
## 17        0.000                  0  0.0780 0.0047        1.0000  11.148  0.2258
## 18        0.000                  0  0.0780 0.0047        1.0000  11.148  0.2258
## 23        1.000                  1  0.1880 0.0025        1.0000   8.024  0.1935
## 24        1.000                  1  0.1880 0.0025        1.0000   8.024  0.1935
## 25        1.000                  1  0.1880 0.0025        1.0000   8.024  0.1935
## 29        1.000                  1  0.1880 0.0025        1.0000   8.024  0.1935
## 34        1.000                  0  0.3722 0.0021        1.0000  12.350  0.1858
## 36        1.000                  0  0.3722 0.0021        1.0000  12.350  0.1858
## 37        1.000                  0  0.3722 0.0021        1.0000  12.350  0.1858
## 38        1.000                  0  0.3722 0.0021        1.0000  12.350  0.1858
## 45        2.000                  1  0.1520 0.0049        1.0000   9.584  0.1935
## 46        2.000                  1  0.1520 0.0049        1.0000   9.584  0.1935
## 47        2.000                  1  0.1520 0.0049        1.0000   9.584  0.1935
## 48        2.000                  1  0.1520 0.0049        1.0000   9.584  0.1935
## 50        2.000                  1  0.1520 0.0049        1.0000   9.584  0.1935
## 52        2.000                  1  0.1520 0.0049        1.0000   9.584  0.1935
## 55        2.000                  1  0.1526 0.0049        1.0000   9.584  0.1937
## 56        2.000                  0  0.1000 0.0058        1.0000  10.846  0.1961
## 57        2.000                  0  0.1000 0.0058        1.0000  10.846  0.1961
## 58        2.000                  0  0.1000 0.0058        1.0000  10.846  0.1961
## 60        2.000                  0  0.1000 0.0058        1.0000  10.846  0.1961
## 61        2.000                  0  0.1000 0.0058        1.0000  10.846  0.1961
## 62        2.000                  0  0.1000 0.0058        1.0000  10.846  0.1961
## 64        2.000                  0  0.1000 0.0058        1.0000  10.846  0.1961
## 66        2.000                  0  0.1008 0.0059        1.0000  10.846  0.1962
## 67        3.000                  1  0.0782 0.0246        1.0000  10.008  0.2061
## 68        3.000                  1  0.0782 0.0246        1.0000  10.008  0.2061
## 69        3.000                  1  0.0782 0.0246        1.0000  10.008  0.2061
## 70        3.000                  1  0.0782 0.0246        1.0000  10.008  0.2061
## 71        3.000                  1  0.0782 0.0246        1.0000  10.008  0.2061
## 72        3.000                  1  0.0782 0.0246        1.0000  10.008  0.2061
## 74        3.000                  1  0.0782 0.0246        1.0000  10.008  0.2061
## 75        3.000                  1  0.0783 0.0247        1.0000  10.008  0.2063
## 76        3.000                  1  0.0791 0.0248        1.0000  10.008  0.2068
## 77        3.000                  1  0.0822 0.0258        1.0000  10.008  0.2082
## 78        3.000                  0  0.0320 0.0264        1.0000  11.026  0.2077
## 79        3.000                  0  0.0320 0.0264        1.0000  11.026  0.2077
## 86        3.000                  0  0.0321 0.0264        1.0000  11.026  0.2078
## 87        3.000                  0  0.0324 0.0266        1.0000  11.026  0.2083
## 88        3.000                  0  0.0319 0.0276        1.0000  11.026  0.2113
## 89        1.394              0.418  1.0000 0.0015        0.9014  22.494      NA
## 90        1.394              0.418  1.0000 0.0015        0.9014  22.494      NA
## 91        1.394              0.418  1.0000 0.0015        0.9014  22.494      NA
## 92        1.394              0.418  1.0000 0.0015        0.9014  22.494      NA
## 93        1.394              0.418  1.0000 0.0015        0.9014  22.494      NA
## 94        1.394              0.418  1.0000 0.0015        0.9014  22.494      NA
## 96        1.394              0.418  1.0000 0.0015        0.9014  22.494      NA
## 97        1.394              0.418  1.0000 0.0015        0.9014  22.494      NA
## 98        1.394              0.418  1.0000 0.0015        0.9026  22.494      NA
## 99        1.394              0.418  1.0000 0.0014        0.9113  22.494      NA

CV vs Undersmoothing (First smoothness order with smaller number of knots)

##  The average lambda of CV-HAL: 0.0022 (= 1 * lambda_CV )
##  The average lambda of globally undersmoothed HAL: 0.0003 (= 0.1612 * lambda_CV )
##  The average number of bases of CV-HAL: 12.0630
##  The average number of bases of globally undersmoothed HAL: 24.0123
##  The average fitting time for CV-HAL: 5.3135 seconds
##  The average fitting time for globally undersmoothed HAL: 7.8475 seconds

grid of scalers

## [1] "proportion of simulations that cannot compute empirical SD:"
##    1.2    1.1      1 0.6952 0.4833  0.336 0.2336 0.1624 0.1129 0.0785 0.0546 
##  0.002  0.002  0.002  0.004  0.004  0.008  0.014  0.026  0.042  0.062  0.084 
## 0.0379 0.0264 0.0183 0.0127 0.0089 0.0062 0.0043  0.003 0.0021 0.0014  0.001 
##  0.106  0.162  0.212  0.302  0.388  0.472  0.544  0.604  0.698  0.754  0.808

vs GAM & Polynomial regression

n = 500

CV vs Undersmoothing (Zero smoothness order with default number of knots)

##  The average lambda of CV-HAL: 0.0031 (= 1 * lambda_CV )  The average lambda of CV-HAL: 0.0031 (= 1 * lambda_CV )
##  The average lambda of globally undersmoothed HAL: 0.0001 (= 0.0209 * lambda_CV )  The average lambda of globally undersmoothed HAL: 0.0001 (= 0.0209 * lambda_CV )
##  The average number of bases of CV-HAL: 16.6580   The average number of bases of CV-HAL: 16.6580   The average number of bases of CV-HAL: 16.6580
##  The average number of bases of globally undersmoothed HAL: 95.8940
##  The average fitting time for CV-HAL: 3.5177 seconds
##  The average fitting time for globally undersmoothed HAL: 6.4772 seconds

Undersmoothed Smoothness-adaptive HAL

##    smooth_order if_n_knots_default sl_pick lambda lambda_scaler n_basis cv_risk
## 1           0.0                  1  0.0000 0.0016        1.0000   8.056  0.3784
## 12          0.0                  0  0.1400 0.0026        1.0000  14.534  0.1627
## 14          0.0                  0  0.1400 0.0026        1.0000  14.534  0.1627
## 18          0.0                  0  0.1400 0.0026        1.0000  14.534  0.1627
## 19          0.0                  0  0.1400 0.0026        1.0000  14.534  0.1627
## 20          0.0                  0  0.1400 0.0026        1.0000  14.534  0.1627
## 22          0.0                  0  0.1400 0.0026        1.0000  14.534  0.1627
## 23          1.0                  1  0.0140 0.0008        1.0000  10.652  0.1727
## 26          1.0                  1  0.0140 0.0008        1.0000  10.652  0.1727
## 27          1.0                  1  0.0140 0.0008        1.0000  10.652  0.1727
## 29          1.0                  1  0.0140 0.0008        1.0000  10.652  0.1727
## 31          1.0                  1  0.0140 0.0008        1.0000  10.652  0.1727
## 32          1.0                  1  0.0140 0.0008        1.0000  10.652  0.1727
## 34          1.0                  0  0.5074 0.0006        1.0000  18.654  0.1507
## 40          1.0                  0  0.5074 0.0006        1.0000  18.654  0.1507
## 42          1.0                  0  0.5074 0.0006        1.0000  18.654  0.1507
## 43          1.0                  0  0.5074 0.0006        1.0000  18.654  0.1507
## 45          2.0                  1  0.1260 0.0012        1.0000  12.054  0.1559
## 46          2.0                  1  0.1260 0.0012        1.0000  12.054  0.1559
## 47          2.0                  1  0.1260 0.0012        1.0000  12.054  0.1559
## 48          2.0                  1  0.1260 0.0012        1.0000  12.054  0.1559
## 49          2.0                  1  0.1260 0.0012        1.0000  12.054  0.1559
## 50          2.0                  1  0.1260 0.0012        1.0000  12.054  0.1559
## 51          2.0                  1  0.1260 0.0012        1.0000  12.054  0.1559
## 52          2.0                  1  0.1260 0.0012        1.0000  12.054  0.1559
## 53          2.0                  1  0.1260 0.0012        1.0000  12.054  0.1559
## 55          2.0                  1  0.1263 0.0012        1.0000  12.054  0.1559
## 56          2.0                  0  0.1980 0.0012        1.0000  15.320  0.1534
## 57          2.0                  0  0.1980 0.0012        1.0000  15.320  0.1534
## 58          2.0                  0  0.1980 0.0012        1.0000  15.320  0.1534
## 59          2.0                  0  0.1980 0.0012        1.0000  15.320  0.1534
## 60          2.0                  0  0.1980 0.0012        1.0000  15.320  0.1534
## 61          2.0                  0  0.1980 0.0012        1.0000  15.320  0.1534
## 62          2.0                  0  0.1980 0.0012        1.0000  15.320  0.1534
## 64          2.0                  0  0.1980 0.0012        1.0000  15.320  0.1534
## 65          2.0                  0  0.1980 0.0012        1.0000  15.320  0.1534
## 67          3.0                  1  0.0040 0.0101        1.0000  11.850  0.1679
## 76          3.0                  1  0.0040 0.0101        1.0000  11.850  0.1679
## 77          3.0                  1  0.0044 0.0102        1.0000  11.850  0.1674
## 78          3.0                  0  0.0040 0.0096        1.0000  14.362  0.1666
## 87          3.0                  0  0.0040 0.0096        1.0000  14.362  0.1665
## 88          3.0                  0  0.0022 0.0096        1.0000  14.362  0.1660
## 89          1.2              0.144  1.0000 0.0006        0.8050  36.472      NA
## 90          1.2              0.144  1.0000 0.0006        0.8050  36.472      NA
## 93          1.2              0.144  1.0000 0.0006        0.8050  36.472      NA
## 97          1.2              0.144  1.0000 0.0006        0.8050  36.472      NA
## 98          1.2              0.144  1.0000 0.0006        0.8075  36.472      NA
## 99          1.2              0.144  1.0000 0.0006        0.8451  36.472      NA

CV vs Undersmoothing (First smoothness order with smaller number of knots)

##  The average lambda of CV-HAL: 0.0006 (= 1 * lambda_CV )
##  The average lambda of globally undersmoothed HAL: 0.0001 (= 0.1644 * lambda_CV )
##  The average number of bases of CV-HAL: 17.7651
##  The average number of bases of globally undersmoothed HAL: 38.3843
##  The average fitting time for CV-HAL: 10.3585 seconds
##  The average fitting time for globally undersmoothed HAL: 20.3527 seconds

grid of scalers

## [1] "proportion of simulations that cannot compute empirical SD:"
##    1.2    1.1      1 0.6952 0.4833  0.336 0.2336 0.1624 0.1129 0.0785 0.0546 
##  0.002  0.002  0.002  0.004  0.000  0.012  0.014  0.030  0.068  0.136  0.196 
## 0.0379 0.0264 0.0183 0.0127 0.0089 0.0062 0.0043  0.003 0.0021 0.0014  0.001 
##  0.262  0.374  0.498  0.592  0.700  0.764  0.836  0.886  0.936  0.934  0.962

vs GAM & Polynomial regression

n = 1000

CV vs Undersmoothing (Zero smoothness order with default number of knots)

##  The average lambda of CV-HAL: 0.0021 (= 1 * lambda_CV )
##  The average lambda of globally undersmoothed HAL: 0.0000 (= 0.0141 * lambda_CV )  The average lambda of globally undersmoothed HAL: 0.0000 (= 0.0141 * lambda_CV )
##  The average number of bases of CV-HAL: 22.4800   The average number of bases of CV-HAL: 22.4800
##  The average number of bases of globally undersmoothed HAL: 195.0103
##  The average fitting time for CV-HAL: 11.6560 seconds
##  The average fitting time for globally undersmoothed HAL: 21.1333 seconds

Undersmoothed Smoothness-adaptive HAL

##    smooth_order if_n_knots_default sl_pick lambda lambda_scaler n_basis cv_risk
## 1         0.000                  1  0.0000 0.0009        1.0000   8.630  0.3744
## 12        0.000                  0  0.0200 0.0017        1.0000  17.136  0.1544
## 13        0.000                  0  0.0200 0.0017        1.0000  17.136  0.1544
## 23        1.000                  1  0.0000 0.0004        1.0000  12.620  0.1632
## 34        1.000                  0  0.8724 0.0002        1.0000  25.174  0.1332
## 35        1.000                  0  0.8724 0.0002        1.0000  25.174  0.1332
## 37        1.000                  0  0.8724 0.0002        1.0000  25.174  0.1332
## 42        1.000                  0  0.8724 0.0002        1.0000  25.174  0.1332
## 45        2.000                  1  0.0020 0.0010        1.0000  12.658  0.1462
## 56        2.000                  0  0.1040 0.0009        1.0000  16.592  0.1406
## 58        2.000                  0  0.1040 0.0009        1.0000  16.592  0.1406
## 59        2.000                  0  0.1040 0.0009        1.0000  16.592  0.1406
## 67        3.000                  1  0.0000 0.0101        1.0000  11.888  0.1607
## 68        3.000                  1  0.0000 0.0101        1.0000  11.888  0.1607
## 77        3.000                  1  0.0000 0.0101        1.0000  11.888  0.1609
## 78        3.000                  0  0.0000 0.0100        1.0000  14.862  0.1582
## 88        3.000                  0  0.0000 0.0100        1.0000  14.862  0.1588
## 89        1.086              0.002  1.0000 0.0001        0.2675  51.896      NA
## 90        1.086              0.002  1.0000 0.0001        0.2675  51.896      NA
## 92        1.086              0.002  1.0000 0.0001        0.2675  51.896      NA
## 93        1.086              0.002  1.0000 0.0001        0.2675  51.896      NA
## 94        1.086              0.002  1.0000 0.0001        0.2675  51.896      NA
## 97        1.086              0.002  1.0000 0.0001        0.2675  51.896      NA
## 99        1.086              0.002  1.0000 0.0001        0.2688  51.896      NA

CV vs Undersmoothing (First smoothness order with smaller number of knots)

##  The average lambda of CV-HAL: 0.0002 (= 1 * lambda_CV )
##  The average lambda of globally undersmoothed HAL: 0.0000 (= 0.1767 * lambda_CV )
##  The average number of bases of CV-HAL: 24.1575
##  The average number of bases of globally undersmoothed HAL: 52.2648   The average number of bases of globally undersmoothed HAL: 52.1975
##  The average fitting time for CV-HAL: 19.1449 seconds
##  The average fitting time for globally undersmoothed HAL: 37.1238 seconds

grid of scalers

## [1] "proportion of simulations that cannot compute empirical SD:"
##    1.2    1.1      1 0.6952 0.4833  0.336 0.2336 0.1624 0.1129 0.0785 0.0546 
##  0.000  0.000  0.000  0.004  0.000  0.008  0.022  0.032  0.080  0.112  0.194 
## 0.0379 0.0264 0.0183 0.0127 0.0089 0.0062 0.0043  0.003 0.0021 0.0014  0.001 
##  0.300  0.402  0.502  0.560  0.644  0.682  0.702  0.768  0.796  0.818  0.826

vs GAM & Polynomial regression